Projective varieties of maximal sectional regularity

TL;DR

This paper classifies projective varieties of maximal sectional regularity, showing they are either divisors on rational normal scrolls or projections of rational normal scrolls, with detailed descriptions under certain conditions.

Contribution

It provides a complete classification of varieties with maximal sectional regularity, identifying their structure as divisors on scrolls or projections thereof.

Findings

Varieties are either divisors on rational normal scrolls or projections of such scrolls.

Explicit classification when the dimension is 2 or the field characteristic is zero.

Characterization of the intersection with a linear subspace as a hypersurface of specific degree.

Abstract

We study projective varieties $X \subset \mathbb{P}^r$ of dimension $n \geq 2$, of codimension $c \geq 3$ and of degree $d \geq c + 3$ that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity $\reg (\mathcal{C})$ of a general linear curve section is equal to $d -c+1$, the maximal possible value (see \cite{GruLPe}). As one of the main results we classify all varieties of maximal sectional regularity. If $X$ is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal $(n+1)$-fold scroll $Y \subset \mathbb{P}^{n+3}$ or else (b) there is an $n$-dimensional linear subspace $\mathbb{F} \subset \mathbb{P}^r$ such that $X \cap \mathbb{F} \subset \mathbb{F}$ is a hypersurface of degree $d-c+1$. Moreover, suppose that $n = 2$ or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of $X$ as a birational linear projection of a rational normal $n$-fold scroll.